INDUCTION John Stuart Mill wrote the first comprehensive study of inductive logic. Deduction had been studied extensively since ancient times, but induction had to wait until the 19 th century! The cartoon above illustrates Mill s dedication to using philosophy to improve the lives of people, but it leaves out one of Mill s most passionate causes full political equality for women! In an inductive argument, the relationship claimed to exist between the truth of the premises and the truth of the conclusion is probability, not certainty. Example: Most Greeks drink wine. Socrates is a Greek. Therefore, probably, Socrates drinks wine In this argument, the truth of the premises could make the truth of the conclusion likely, but not certain. All inductive reasoning is based on an assumption called the UNIFORMITY OF NATURE. This principle stipulates that well-established patterns observed in the past will persist in the present and future. Therefore, the past can be used to predict what will happen in the near and remote future. Without this assumption, it would be impossible to learn from experience, and therefore neither science nor common sense would be possible. The conclusion of an inductive argument should normally be modified by such words as probably, or it is likely. But sometimes we treat highly probable conclusions as if they were certain. For example, your conclusion that a speeding bus would kill you if you stepped in front of it, is based on patterns observed in the past, and therefore inductive. But would you normally feel compelled to treat it merely as a probable conclusion?
RELATIONSHIP BETWEEN PREMISES & CONCLUSION The logical form of an inductive argument may help illustrate what the argument is trying to accomplish, but it will not show whether the premises and conclusion are properly connected. If the truth of the premises would succeed in making the truth of the conclusion probable in the manner claimed, the relationship is called STRONG (not valid. ). Remember, induction is an attempt to apply the idea of the Uniformity of Nature. Therefore, the common sense rule of induction is that we want as close a match as possible between the evidence we present in the premises, and what we predict in the conclusion. All other criteria for judging inductive arguments are an elaboration of this rule. Here is a summary of the differences between deductive validity and inductive strength. DEDUCTIVE VALIDITY Relationship of certainty All or nothing Logical form is crucial Content is irrelevant INDUCTIVE STRENGTH Relationship of probability Matter of degree Logical form is not decisive Content is crucial CONTRAST DEDUCTION AND INDUCTION Consider this valid, deductive argument. All people are mortal. Socrates is a person. Therefore, Socrates is mortal. The diagram illustrates that if everything in the set of all people has the property of being mortal, and Socrates is in that set, then he must have that property. We don t need to ADD anything to the diagram, once the premises have been represented. The conclusion is already logically contained by the premises.
This is NEVER the case for an inductive argument. We say the conclusion of an inductive argument runs ahead of the premises. In the premises, we are stating the consistent patterns we have observed. In the conclusion, we are predicting things we have not yet observed. Obviously, we want our prediction to be as good a match as possible the patterns we have observed. The closer the match, the stronger the argument. The more flimsy the match, the weaker the argument. Example ~ When a meteorologist predicts a 30% chance of rain, s/he is saying that in 30% of the times we have observed conditions like those we have today, rain has also occurred. Is rain a certainty? No. Is there a likelihood, based on past experience? Yes. TYPES OF INDUCTIVE ARGUMENTS There are many flavors of inductive arguments, but we will examine only two very basic types. INDUCTIVE GENERALIZATION moves from an observation of some members of a set (a sample) to a prediction about the entire set (a population). Two Examples ~ All the swans I observed were white. I have heard of no exceptions. Therefore, I conclude that probably all swans are white. Most of the swans I observed were white. I have heard of few exceptions. Therefore, I conclude that probably most swans are white. Logical Form = All (or most) observed X have property Y. No (or few) exceptions are known Therefore, (probably) All (or most) X have property Y.
Some definitions ~ Generalization = a conclusion about an entire group. Population = the group about which you are generalizing. Sample = the subset of the population you have actually been able to observe. INDUCTIVE ANALOGY An analogy is a comparison. Inductive analogies move: from an observation that an individual resembles some members of a set in regard to particular properties, to a prediction that the individual will resemble those same members as regards other properties. Example ~ Emily, Adrienne, Iris, Alicia, Tori are all swans, all bug-eaters, all female, all white, all migratory. Jennifer is a swan, eats bugs, is female, is white. Therefore, I predict that Jennifer is probably migratory. Logical Form = A, B, C, D, E, all have properties V, W, X, Y, Z. N has properties V, W, X, Y Therefore (probably) N also has property Z.
NOTE: The easiest way to distinguish these two forms is to look at the conclusion. In an inductive generalization, the conclusion will be about an entire population. In an inductive analogy, the conclusion will be about an individual. CHECK YOURSELF 1. Why do we never use the words valid or sound to describe inductive arguments? 2. What is the principle of the uniformity of nature? 3. What do we mean when we say an inductive argument is strong or weak? Answers: 1. Validity is a relationship of certainty between the premises and conclusion. In inductive arguments, we never have certainty. To be sound, an argument must be valid and have all true premises. Since inductive arguments cannot be valid, they cannot be sound. 2. Well-established patterns observed in the past will persist in the present and future. Therefore, we can learn from experience in a way that allows us to make reasonable (not certain) predictions about the present and the future. 3. Strong and weak refer to the relationship between the premises and conclusion in an inductive argument. An inductive argument is strong if we have drawn an appropriate conclusion based on our premises, and there is a consistency between what we observed and what we predict. CONFIDENT? READ ON. NOT CONFIDENT? REVIEW!
EVALUATING INDUCTIVE ARGUMENTS - SIX CRITERIA Clearly, inductive reasoning can go wrong. We need criteria for evaluating inductive arguments. We cannot use the same criteria we use for deduction, since we are looking for a different kind of relationship. This next section provides a checklist for evaluating inductive reasoning. CHECK LIST FOR INDUCTIVE ARGUMENTS 1. Are the premises true? We normally do not care what follows from false premises, whether we are reasoning deductively or inductively. We need to check the facts, do the math, and get the premises right to the best of our ability. 2. How broad is the sample? The variety in the sample should be a good match for the variety in the population. How narrow is the sample? A sample may be broad in some ways, but narrow in others. Suppose I was given a research grant to study the plumage of all swans. I do my research, and conclude: All the swans I observed were white. I have heard of no exceptions. Therefore, I conclude that probably all swans are white. Now suppose I realize that ALL the swans I observed (my entire sample) were male, 1 year old, North American, swamp-dwelling, and vegetarian. Oops.
My population all swans may contain female or ungendered swans, many age groups, swans from other continents and other habitats, and who eat different diets. Any of this could potentially affect the color of their plumage. My sample in NOT a good match for the population, and so I do not have a strong argument. Applying the Principle of the Uniformity of Nature, I have observed patterns only in a sub-group (male, 1 year old, North American, swamp-dwelling, and vegetarian.) If I try to draw conclusions about ALL swans, my conclusion is running miles ahead of my evidence! Now suppose I do additional research, and include in my sample both male and female swans, and swans of different ages. Is the argument better than it was before? Yes. Does it still have serious weaknesses? Yes. Remember, inductive strength is a matter of degree, and the same argument may be strong in some ways, but weak in others. 3. How big is the sample? Increasing sample size may help if the added numbers make the variety in the sample a better match for the variety in the population. Suppose I have only observed 100 swans. If I go out and observe 1,000 more swans, that may mean I have a better chance of getting a sample that has the same variety as the population. It is not quite that simple, though. What if observe 1,000 more swans, but the 1,000 additional swans are all male, 1 year old, North American, swamp-dwelling, and vegetarian? My sample is bigger, but not better! 4. How sweeping and confident is the conclusion? The more sweeping or confident, the better the evidence needs to be. Suppose I am unable to do any more research on swans. Since I cannot add to my observations, I can alter my conclusion to make it a better match, by making it less sweeping or less confident. Sweeping = how big a group do we include? If I restricted my conclusion to the plumage of male, 1 year old, North American, swamp-dwelling, vegetarian swans (instead of ALL swans,) my sample would be a much better match! Confident = how strong a connection do I claim exists between my evidence and my conclusion. Instead of concluding Probably, all swans are white, I could conclude something like We have some initial evidence to suspect swans may generally be white, but more research is needed. The tentative, less confident wording of the conclusion makes the conclusion a better match for the actual evidence. 5. Have we made a conscientious effort to examine all the relevant evidence? Unless we are omniscient, we cannot be sure we have examined all the relevant evidence. (Of course, if we were omniscient, why would we need induction?)
We can, however, make a conscientious effort. If someone making an inductive argument has clearly omitted important sources of evidence, the premises become suspect! For example, did I look at scientific journals, and reputable scientific web sites, to see what others had observed? If not, I may have missed important data! No, actually I have never met Natalie Portman. 6. For an analogy, have looked at the whole pattern, weighing significant similarities against significant differences? Earlier, we used this example of an inductive analogy. Emily, Adrienne, Iris, Alicia, Tori are all swans, all bug-eaters, all female, all white, all migratory. Jennifer is a swan, eats bugs, is female, is white. Therefore, I predict that Jennifer is probably migratory. We saw a pattern of similarity between the individual (Jennifer) and the group described in the first premise. What if we also saw a pattern of differences? For example ~ What if Jennifer is an Australian swan, and the others are North American swans? What if Jennifer is 10 years old, and the others are all 2 years old? What if the others live in a flock, and Jennifer is a solitary swan? If the pattern of difference is as strong, or stronger than the pattern of similarity, then the Principle of the Uniformity of Nature does not support predicting Jennifer is migratory. You must weigh the pattern of similarity against the pattern of differences to draw your conclusion. READY FOR PRACTICE EXERCISES? PART ONE: Answer TRUE or FALSE, and give a FULL EXPLANATION of your answer. Define important terms as part of your explanation. 1. If an inductive argument is strong, it can still have a false conclusion.
2. A good inductive argument must have true premises, and be valid. 3. If an inductive argument has true premises and a true conclusion, then it is strong. ANSWERS ~ 1. True. In induction, the truth of the conclusion is NEVER guaranteed. 2. False. We do want true premises, but if the form is valid, then by definition we are looking at a deductive argument, not an inductive argument. 3. False. Strong refers to the relationship between the premises and conclusion. You may have true premises and a true conclusion, but they may not be properly related to one another. PART TWO: Choose the one best answer, and defend your choice. Most of the Hindus I have known have been vegetarians. Gupta is a Hindu. I d bet he s a vegetarian. 1. This argument is an example of: a. Inductive generalization. b. Inductive analogy. 2. If we conducted our observations only in vegetarian restaurants, that would make our conclusion: a. Weaker, because the sample is more narrow. b. Weaker, because the sample is more broad. c. Stronger, because the sample is more narrow. d. Stronger, because the sample is more broad. 3. If all the Hindus we observed to be vegetarians also came from the same area as Gupta and worshipped at the same temple as Gupta, that would make the argument: a. Weaker, because the sample has more in common with Gupta. b. Weaker, because the sample is more narrow. c. Stronger, because the sample has more in common with Gupta. d. Stronger, because the sample is more broad.
ANSWERS ~ 1. b, inductive analogy. The conclusion is about an individual (Gupta,) not a group. 2. a, weaker because the sample is too narrow. You are likely to find vegetarians at a vegetarian restaurant, regardless of whether they are Hindu, Catholic, Atheist or whatever. 3. c, stronger because the sample has more in common with Gupta. Remember, in an inductive analogy, the greater the pattern of resemblance, the stronger the argument. CONFIDENT? IF YES, THEN DO THE EXERCISES BELOW. For #1 & #2, answer true or false and explain your answer. Be sure to define key terms in your explanation. 1. Making the sample bigger will always make an inductive generalization stronger. 2. A strong inductive argument must have a highly probable conclusion. For # 3 - #5, choose the one best answer, and explain your answer 3. Most of the students, who I know personally, refrain from partying before exams. I know many students. Yusif is also a student. Therefore, I d guess that probably Yusif also refrains from partying before exams.
This argument above (#3) is an example of: a) invalid deductive reasoning. b) inductive analogy. c) inductive generalization.. 4. Most of the students, who I know personally, refrain from partying before exams. I know many students. Yusif is also a student. Therefore, I d guess that probably Yusif also refrains from partying before exams. If most of the students I know were from Chicago, that would make the argument: a) stronger, because the sample is more narrow. b) weaker, because the sample is more narrow. c) stronger, because the sample is more broad. d) weaker, because the sample is more broad. 5. Most of the students I know personally refrain from partying before exams. I know many students. Yusif is also a student. Therefore, I d guess that probably Yusif also refrains from partying before exams. Suppose most of the students I know were from Chicago, were Muslims, and were younger than 25 years old. If Yusif is also from Chicago, a Muslim, and younger than 25 years old, that would make the argument: a) weaker because the sample is smaller. b) weaker, because the sample is like Yusif. c) stronger, because the sample is more like Yusif. d) stronger, because the sample is more broad. 6. Most of the students, who I know personally, including Yusif, refrain from partying before exams. I know many students. Therefore, I d guess that probably most students refrain from partying before exams. This argument above (#6) is an example of: d) invalid deductive reasoning. e) inductive analogy. f) inductive generalization.